Questions tagged [semi-riemannian-geometry]
Manifolds with a non-degenerate symmetric bilinear form in each tangent space varying differentiably but with constant index and signature.
26
questions with no upvoted or accepted answers
7
votes
0
answers
231
views
Completeness is a conformal invariant
In this article about completeness of semi-riemannian manifolds the author writes (in section 2.2) that it is unkown if the following statement holds:
A compact indefinite manifold which is conformal ...
- 893
6
votes
0
answers
516
views
Time-separation function on "globally hyperbolic" spacetimes with everywhere timelike boundary
It is well-known that, in globally hyperbolic spacetimes, the time separation function $\tau$ (aka Lorentzian distance function) enjoys the following property: fix a point $p$ and a point $q \in I^-(p)...
- 473
5
votes
0
answers
113
views
Compact, incomplete semi-Riemannian manifold of constant curvature
In the Riemannian setting, Hopf-Rinow tells us that any compact Riemannian manifold is complete. The Clifton-Pohl torus gives a counter example for indefinite metrics.
However, in the Lorentz setting,...
- 73
5
votes
0
answers
324
views
"Correct" definition of signed curvature in Minkowski plane
We know that for $n\geq 2$ the de Sitter space $\mathbb{S}^n_1(r)$ and the hyperbolic space $\mathbb{H}^n(r)$ have constant curvature $1/r^2$ and $-1/r^2$, respectively.
Looking at references such as ...
- 1,133
4
votes
0
answers
135
views
Examples of indefinite Einstein non-Ricci-flat metric on solvmanifolds
A solvmanifold (resp. nilmanifold) is a compact quotient of a solvable (risp. nilpotent) Lie group $G$. Usually one consider a co-compact lactice $\Gamma$ on $G$ and the solvmanifold is $G/\Gamma$. ...
4
votes
0
answers
99
views
Positive and non-negative sectional curvature of semi-riemannian metrics
I am currently studying the techniques related to geometry of positive and non-negative sectional curvature of Riemannian metrics. In particular, I have done some work with Cheeger deformations. I ...
- 1,774
3
votes
0
answers
59
views
Searching for a type of geometric flow in Lorentzian geometry
Let $(N,g)$ be a globally hyperbolic Lorentzian manifold. Given any smooth hypersurface $\Sigma$ in $(N,g)$ we define $\|\Sigma\|= \sup_{p \in N,X \in T_p\Sigma} |h(X,X)|$ where $h$ is the second ...
- 4,111
3
votes
0
answers
118
views
Lorentzian manifolds of negative spacelike sectional curvature
Suppose $(M,g)$ is a simply connected Lorentzian manifold of signature $(-,+,\ldots,+)$ and such that the sectional curvature of any space-like surface is non-positive. Is it true that there are no ...
- 4,111
3
votes
0
answers
320
views
Maximal symmetry at the speed of light
Are there examples of 1 + 3 dimensional pseudo-Riemannian manifolds with 6 dimensional isometry group whose orbits are light-like (i.e., the metric restricted to each orbit is degenerate)?
Here is a (...
- 427
3
votes
0
answers
161
views
Parallel frame for marginally trapped bi-harmonic surfaces in $\Bbb R^4_2$
I'm reading the paper Classification of marginally trapped Lorentzian flat surfaces in $\mathbb{E}^4_2$ and its applications to biharmonic surfaces by B. Y. Chen.
Summarizing it quickly: he first ...
- 1,133
2
votes
0
answers
76
views
Examples of curvature-adapted subgroups of semi-Riemannian groups
Let $G$ be a semi-Riemannian group, i.e., a Lie group equipped with a bi-invariant semi-Riemannian metric. I am looking for examples of Lie subgroups that are curvature adapted to $G$.
First, allow me ...
- 935
2
votes
0
answers
46
views
On certain umbilic surfaces
Let $(M,g)$ be a three dimensional Lorentzian manifold with signature $(-,+,+)$ and let $p\in M$ and let $U$ be a small neighborhood of $p$. Suppose there is a smooth timelike surface $S$ containing $...
- 4,111
2
votes
0
answers
77
views
Analogous $H^1$-space for pseudo inner products
Perhaps this is a naive question but I could not find anything related to this.
Imagine we are on a bounded and regular open subset $\Omega$ of $\mathbb{R}^3_1$, i.e, $\mathbb{R}^4$ is considered ...
- 1,774
2
votes
0
answers
273
views
Compatibility of Kirillov-Kostant-Souriau form and Killing form
Let $\mathfrak{g}$ be a real semisimple Lie algebra. We know that a coadjoint orbit $\mathcal{O} \hookrightarrow \mathfrak{g}^*$ carries a natural symplectic form $\omega$, namely the Kirillov-Kostant-...
- 461
2
votes
0
answers
166
views
Reference for connection of a Hessian metric
Let $(M,\langle \cdot,\cdot\rangle)$ be a pseudo-Riemannian manifold and $f: M \to \Bbb R$ be a smooth function. One can consider the covariant Hessian $\nabla ({\rm d}f)$. Some time ago I had seen a ...
- 1,133
1
vote
0
answers
90
views
"Classifying" causally closed sets in Minkowski space
Let $M = \mathbb R^{D+1}$ be Minkowski space. Recall that the causal complement of a set $A \subseteq M$ is the set $A^\perp \subseteq M$ where $p \in A^\perp$ there is no timelike path between $p$ ...
- 61.6k
1
vote
0
answers
64
views
A question on future Cauchy developement
Let us consider the Minkowski spacetime $\mathbb R^{1+2}$ equipped with the metric
$$ \eta(t,x) = -(dt)^2+ (dx^1)^2+(dx^2)^2.$$
Let $\Omega$ be a bounded simply connected domain in $\mathbb R^2$ with ...
- 4,111
1
vote
0
answers
71
views
Completeness of infinitely intersecting causal geodesics in strongly causal spacetimes
Let $(M,g)$ be a connected, smooth, strongly causal Lorentzian manifold, and consider an inextendible causal geodesic $\sigma : [0,b) \to M$ (a priori, $b$ may be $\infty$) with the following property:...
- 111
1
vote
0
answers
73
views
Smooth closed Riemannian manifolds with quasi-analytic metrics
I haven't found a reference for this exact definition in the literature so I wonder if what I want to pose here makes sense. I basically want to consider a smooth close Riemannian manifold that ...
- 4,111
1
vote
0
answers
300
views
Conformal changes of metric and Ricci curvature
Let $(M,g)$ be a three dimensional smooth Lorentzian manifold and let $p$ be a fixed point in $M$ and let $S$ be a smooth symmetric tensor of rank two on $T_pM\times T_pM$. Does there exist a smooth ...
- 4,111
1
vote
1
answer
228
views
On intersection of null geodesics
Let us consider a globally hyperbolic Lorentzian manifold $(M,g)$ with empty cut locus. Suppose that
$p$ is a point in $M$ and consider $C^-(p)$ to be the past null cone in $M$ emanating from the ...
- 4,111
1
vote
0
answers
181
views
Surjectivity of Pseudo-Riemannian exponential map on geodesically complete manifolds
Suppose one has a geodesically complete pseudo-Riemannian manifold $M$ i.e. the exponential map is defined for all tangent vectors on the manifold. Can one make a sensible statement about whether (or ...
- 611
0
votes
0
answers
102
views
Lorentzian geometry. Comparing Honda's main theorem A construction to mine: Mixed type surfaces
This question is based on a wonderful paper by A. Honda (link below) where his main theorem A provides an incredible uniqueness result. Mixed type surfaces and type changing metrics have been ...
- 171
0
votes
0
answers
185
views
Can the Lie derivative of a Riemannian metric be expressed in terms of the Lie derivative of a Lorentzian metric?
On a Lorentzian manifold with metric (M,g) with a vanishing Euler-Poincare characteristic, there exists a line element vector X which has a collinear vector u (Manifold Theory: An introduction for ...
- 9
0
votes
0
answers
65
views
What is the meaning and role of global hyperbolicity condition for semi-Riemannian manifolds
What is the heuristic meaning of the global hyperbolicity condition for semi-Riemannian manifolds?
Also what is the role of this condition in the study of geodesic connectedness?
user60665
0
votes
0
answers
138
views
(Semi-)Riemannian geometry for working PDE analysts
What is a good reference on (semi-)Riemannian geometry written for PDE analysts (that is, with main focus on analytical problems and approaches)?
The closest thing I know to this, are two books by ...
user60665